Article history: Received 31 January 2008 Received in revised form 18 June 2008 Accepted 23 July 2008 Available online 19 August 2008

High order energy-preserving methods for Hamiltonian systems are presented. For this aim, an energy-preserving condition of continuous stage Runge–Kutta methods is proved. Order conditions are simplified and parallelizable conditions are also given. The computational cost of our high order methods is comparable to that of the average vector field method of order two.

In this paper we introduce and study rational slice monogenic functions. After proving a decomposition theorem for such functions, we are able to prove the Runge approximation theorem for slice monogenic functions. We then show how a similar argument can be used to obtain an analogue of the Runge approximation theorem in the slice regular setting.

The main purpose of this paper is to discuss the equilibrium attractive properties of a class of multistep Runge–Kutta methods for initial value problems of ordinary differential equations. Some algebraic conditions insuring the equilibrium attractivity are given, and some methods satisfying these algebraic conditions are constructed. Some numerical examples confirm our results. 2005 Elsevier I...

Abstract In this work, a version of continuous stage stochastic Runge–Kutta (CSSRK) methods is developed for differential equations (SDEs). First, general order theory these established by the B-series and multicolored rooted tree. Then proposed CSSRK are applied to three special kinds SDEs corresponding conditions derived. particular, single integrand with additive noise, we construct some spe...

A personal view of the historical development of software based on explicit Runge-Kutta formulas is presented, and some future developments are suggested.

This work constructs a new class of multirate schemes based on the recently developed generalized additive Runge-Kutta (GARK) methods [10]. Multirate schemes use different step sizes for different components and for different partitions of the right-hand side based on the local activity levels. We show that the new multirate GARK family includes many well-known multirate schemes as special case...